Algebra I. Homework 1. Due on September 17, 2020.
1. Prove or disprove: Z/2Z×Z/2Z'Z/4Z.
2. Assume that the elements a, b, c of a group G satisfy abc =e, where e is the neutral element of G.
(a) does this imply that bca =e?
(b) does this imply that bac =e?
Give a proof if the answer is positive, give a counterexample if the answer is negative.
3. Determine the number of elements of order 2 in the symmetric group S4. 4. Classify groups of order 6by analyzing the following three cases:
(a) G contains an element of order 6.
(b) G contains an element of order 3 but not of order 6.
(c) All elements of G have order 1 or2.
5. Quaternions. Denote by H8 the subgroup ofGL2(C), generated by I =
0 1
−1 0
, J =
0 i
i 0
, K =
i 0
0 −i
.
Compute the order of H8, find all subgroups of H8, find normal subgroups of H8 and the corresponding quotient groups.
6. Give an example of a finite group G that satisfies the following properties:
(a) G is of order 8;
(b) G is not abelian;
(c) all the subgroups of G are normal.
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